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- W2960945206 abstract "A metric measure space $(X,d,mu)$ is said to be $A_{infty}$ on curves if there exist constants $tau 0$ with the following property. For every $x in X$, $0 tau mu(B(x,r))$, there exists a continuum $gamma subset X$ of length $leq r$ satisfying $mathcal{H}^{1}_{infty}(gamma cap S) geq theta r$. I first observe that spaces of $Q$-bounded geometry, $Q > 1$, are $A_{infty}$ on curves. Then, I show that any complete, doubling, and quasiconvex space $(X,d,mu)$ which is $A_{infty}$ on curves has Alberti representations with $L^{p}$-densities for some $p > 1$, depending only on the doubling and $A_{infty}$-constants. More precisely, any normalised restriction of $mu$ to a ball $B subset X$ can be written as $mu_{B} = f_{B} , dnu_{B}$, where $nu_{B}$ is a convex combination of measures of linear growth supported on continua of length $le mathrm{diam}(B)$, and $|f_{B}|_{L^{p}(nu_{B})} leq C$ for some constant $C geq 1$ independent of $B$." @default.
- W2960945206 created "2019-07-23" @default.
- W2960945206 creator A5091472615 @default.
- W2960945206 date "2019-07-16" @default.
- W2960945206 modified "2023-09-27" @default.
- W2960945206 title "Quantitative Alberti representations in spaces of bounded geometry" @default.
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